The Stacks project

Proof. The equivalence (1) $\Leftrightarrow$ (2) follows from the definitions and Morphisms, Lemma 29.13.3. For (2) $\Rightarrow$ (4), let $U$ and $V$ be as in (4). By Schemes, Lemma 26.21.14, the induced morphism $U \to V$ is quasi-compact. Thus, for every affine open $V' \subset V$, the fiber product $V' \times _ V U$ is quasi-compact, so, by (2), the induced map $V' \times _ V U \to V'$ is quasi-affine. Thus, $U \to V$ is also quasi-affine by Morphisms, Lemma 29.13.3. This argument also gives (3) $\Rightarrow$ (4): indeed, keeping the same notation, those affine opens $V' \subset V$ that lie in one of the $V_ j$ cover $V$, so one needs to argue that the quasi-compact map $V' \times _ V U \to V'$ is quasi-affine. However, by (3), the composition $V' \times _ V U \to V' \to V_ j$ is quasi-affine and, by Schemes, Lemma 26.21.13, the map $V' \to V_ j$ is quasi-separated. Thus, $V' \times _ V U \to V'$ is quasi-affine by Morphisms, Lemma 29.13.8. The final implications (4) $\Rightarrow$ (2) and (4) $\Rightarrow$ (3) are evident. $\square$

Proof. Let $f : X \to Y$ and $g : Y \to Z$ be ind-quasi-affine morphisms. Let $V \subset Z$ and $U \subset f^{-1}(g^{-1}(V))$ be quasi-compact opens such that $V$ is also quasi-separated. The image $f(U)$ is a quasi-compact subset of $g^{-1}(V)$, so it is contained in some quasi-compact open $W \subset g^{-1}(V)$ (a union of finitely many affines). We obtain a factorization $U \to W \to V$. The map $W \to V$ is quasi-affine by Lemma 37.66.2, so, in particular, $W$ is quasi-separated. Then, by Lemma 37.66.2 again, $U \to W$ is quasi-affine as well. Consequently, by Morphisms, Lemma 29.13.4, the composition $U \to V$ is also quasi-affine, and it remains to apply Lemma 37.66.2 once more. $\square$

Proof. The first assertion is immediate from the definitions. In particular, affine morphisms, such as closed immersions, are ind-quasi-affine. Thus, by Lemma 37.66.3, it remains to show that an open immersion is ind-quasi-affine. This, however, is immediate from the definitions. $\square$

Proof. By Lemma 37.66.2, we may work Zariski locally on $Z$ and then on $Y$, so we lose no generality by assuming that $Z$, and then also $Y$, is affine. Then any quasi-compact open of $X$ is quasi-affine, so Lemma 37.66.2 gives the claim. $\square$

Proof. Let $f : X \to Y$ be an ind-quasi-affine morphism. For checking that every base change of $f$ is ind-quasi-affine, by Lemma 37.66.2, we may work Zariski locally on $Y$, so we assume that $Y$ is affine. Furthermore, we may also assume that in the base change morphism $Z \to Y$ the scheme $Z$ is affine, too. The base change $X \times _ Y Z \to X$ is an affine morphism, so, by Lemmas 37.66.3 and 37.66.4, the map $X \times _ Y Z \to Y$ is ind-quasi-affine. Then, by Lemma 37.66.5, the base change $X \times _ Y Z \to Z$ is ind-quasi-affine, as desired. $\square$

Proof. The stability of ind-quasi-affineness under base change supplied by Lemma 37.66.6 gives one direction. For the other, let $f : X \to Y$ be a morphism of schemes and let $\{ g_ i : Y_ i \to Y\}$ be an fpqc covering such that the base change $f_ i : X_ i \to Y_ i$ is ind-quasi-affine for all $i$. We need to show $f$ is ind-quasi-affine.

By Lemma 37.66.2, we may work Zariski locally on $Y$, so we assume that $Y$ is affine. Then we use stability under base change ensured by Lemma 37.66.6 to refine the cover and assume that it is given by a single affine, faithfully flat morphism $g : Y' \to Y$. For any quasi-compact open $U \subset X$, its $Y'$-base change $U \times _ Y Y' \subset X \times _ Y Y'$ is also quasi-compact. It remains to observe that, by Descent, Lemma 35.23.20, the map $U \to Y$ is quasi-affine if and only if so is $U \times _ Y Y' \to Y'$. $\square$

Proof. Let $f : X \to Y$ be a separated locally quasi-finite morphism of schemes. Let $V \subset Y$ be affine and $U \subset f^{-1}(V)$ quasi-compact open. We have to show $U$ is quasi-affine. Since $U \to V$ is a separated quasi-finite morphism of schemes, this follows from Zariski's Main Theorem. See Lemma 37.43.2. $\square$